Theorem hfmmvalt 9510

Description: Value of the scalar product with a Hilbert space functional.

Assertion

Ref	Expression
hfmmvalt	|- ((A e. CC /\ T:H~-->CC) -> (A .fn T) = {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))})

Distinct variable groups:   x,y,A   x,T,y

Proof of Theorem hfmmvalt

Step	Hyp	Ref	Expression
1		ax-hilex 8864	. . . 4 |- H~ e. V
2	1	opabex2 3616	. . 3 |- {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))} e. V
3		opreq1 3974	. . . . . 6 |- (f = A -> (f x. (g` x)) = (A x. (g` x)))
4	3	eqeq2d 1489	. . . . 5 |- (f = A -> (y = (f x. (g` x)) <-> y = (A x. (g` x))))
5	4	anbi2d 618	. . . 4 |- (f = A -> ((x e. H~ /\ y = (f x. (g` x))) <-> (x e. H~ /\ y = (A x. (g` x)))))
6	5	opabbidv 2675	. . 3 |- (f = A -> {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))} = {<.x, y>. | (x e. H~ /\ y = (A x. (g` x)))})
7		fveq1 3729	. . . . . . 7 |- (g = T -> (g` x) = (T` x))
8	7	opreq2d 3982	. . . . . 6 |- (g = T -> (A x. (g` x)) = (A x. (T` x)))
9	8	eqeq2d 1489	. . . . 5 |- (g = T -> (y = (A x. (g` x)) <-> y = (A x. (T` x))))
10	9	anbi2d 618	. . . 4 |- (g = T -> ((x e. H~ /\ y = (A x. (g` x))) <-> (x e. H~ /\ y = (A x. (T` x)))))
11	10	opabbidv 2675	. . 3 |- (g = T -> {<.x, y>. | (x e. H~ /\ y = (A x. (g` x)))} = {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))})
12		df-hfmul 9505	. . . 4 |- .fn = {<.<.f, g>., h>. | ((f e. CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))})}
13		axcnex 5279	. . . . . . . 8 |- CC e. V
14	13, 1	elmap 4340	. . . . . . 7 |- (g e. (CC ^m H~) <-> g:H~-->CC)
15	14	anbi2i 482	. . . . . 6 |- ((f e. CC /\ g e. (CC ^m H~)) <-> (f e. CC /\ g:H~-->CC))
16	15	anbi1i 483	. . . . 5 |- (((f e. CC /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))}) <-> ((f e. CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))}))
17	16	oprabbii 4003	. . . 4 |- {<.<.f, g>., h>. | ((f e. CC /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))})} = {<.<.f, g>., h>. | ((f e. CC /\ g:H~-->CC) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))})}
18	12, 17	eqtr4 1501	. . 3 |- .fn = {<.<.f, g>., h>. | ((f e. CC /\ g e. (CC ^m H~)) /\ h = {<.x, y>. | (x e. H~ /\ y = (f x. (g` x)))})}
19	2, 6, 11, 18	oprabval2 4034	. 2 |- ((A e. CC /\ T e. (CC ^m H~)) -> (A .fn T) = {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))})
20	13, 1	elmap 4340	. 2 |- (T e. (CC ^m H~) <-> T:H~-->CC)
21	19, 20	sylan2br 455	1 |- ((A e. CC /\ T:H~-->CC) -> (A .fn T) = {<.x, y>. | (x e. H~ /\ y = (A x. (T` x)))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  {copab 2671  -->wf 3184  ` cfv 3188  (class class class)co 3969  {copab2 3970   ^m cm 4328  CCcc 5244   x. cmul 5251  H~chil 8783   .fn chft 8806

This theorem is referenced by:  hfmvalt 9517  brafnmult 9870  kbass2t 10045

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-qs 4272  df-map 4330  df-ni 5012  df-nq 5050  df-np 5098  df-nr 5179  df-c 5252  df-hfmul 9505

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